Lecture on Propositional Logic

MODR 1760 A & E 2024-25 Lecture on Propositional Logic: Truth-Tables: Equivalence, Validity, and Invalidity Last week we defined deductive validity as a feature of arguments that is present when the truth of the premises guarantees the truth of the conclusion of an argument. I explained how a sentence or an argument may have a “logical” form, that is a structure that is given by the use of logical words in that sentence or argument. In this lecture we want to use logical form to explain the equivalence of two logical forms of sentences, and as well to identify valid logical forms. To do this you must learn what a truth-table is. These notes are in agreement with, and add a bit to Govier’s Chapter 8 section, “Testing for Validity by the Truth-Table technique. pp. 222-4. A. Truth-Tables Here is a truth table for the propositional logic sentence ~(A&B). A T T F F B T F T F A&B T F F F ~(A&B) F T T T The first two columns say what value the basic sentences A and B have. Then there is a break because then we insert the sentences that we are interested in knowing the values of. Since ~(A&B) is made up of the component A&B, and A&B is made up for A,B. We are writing the sentences from left to right in increasing complexity. The first two columns and the four rows underneath them lay out the truth-possibilities for the sentence form ~(A&B). This means: Each of the rows is a possible value of the sentences A and B. The first one is the possibility that both A, B are true. The second one makes A true and B false. The third the reverse, and the fourth makes both false. And since we are considering each sentence to be true (T) or false (F). That is all there is. The third column tells us, for each of these truth possibilities, whether A&B is true or false for those values of A and B. This information was given in our definition of the meaning of “&” in the last lecture. We figure out the values of the complex sentences on the basis of the simpler ones using those definitions. The fourth column tells us the value of the sentence, ~(A&B) for each of the truth possibilities. OK, cool, but what use is this? B. Equivalence of Sentences of Prop Logic Each sentence of propositional logic is a logical form an English sentence might have. And it might be useful to know when two propositional logic sentences always got the same values. If they did, they would mean the same thing. To compare their values, we compare their truth tables. Definition: Two sentences of Prop Logic are equivalent means for every truth-possibility of their basic sentences, they get the same truth value. To know if ~A V ~B is logically equivalent to ~(A & B), we could compare their truth-tables. A T T F F B T F T F ~A F F T T ~B ~A V ~B F F T T F T T T A T T F F B A&B ~(A&B) T T F F F T T F T F F T We find that they are equivalent, because the value for each of the rows is the same. I will put the two tables together here and you can see this. A T T F F B A&B ~(A&B) T T F F F T T F T F F T ~A F F T T ~B ~A V ~B F F T T F T T T For each truth-possibility, the sentences ~(A&B) gets the same value as the ~A V ~B. This shows itself in English by the fact that when you say, “Not both A and B” as in “I don’t have apples and bananas.” you say the same things as “Either ~A or not B” as in “I don’t have apples or I don’t have bananas.” Logically equivalences are useful to be aware of. “Neither A nor B” ~(AVB). is the same as “~A and ~B.” Both not-A and not-B. These two equivalences are called DeMorgan’s Laws. C. Deductive Validity of Logical Forms in Propositional Logic Definition of deductive validity for propositional logic: An argument consisting only of propositional logic sentences is deductively valid means, every truth possibility that makes the premises of the argument true, also makes the conclusion true. (Save yourself trouble: Memorize this definition!) For logical forms, we will learn how to demonstrate deductive validity, and it means we can prove the validity of a logical form by constructing a truth-table that uses all the basic statements of that argument. D. General Truth-Table Method To test for the validity of a bunch of propositional logic sentences, we need a general method for constructing a truth-table. That method depends on how many basic sentence letters the complex sentence (or argument) contains. 1. Note the number of atomic sentence letters (or basic sentence letters) in the Prop Logic sentence (or argument) you are building a truth table for. Make columns on the left for each atomic sentence. Example 1: Construct a truth table for the Prop Logic argument A -> B ~A Therefore, ~B. How many single statement letters are in this argument? There are 2, A and B. The general rules is: there will be 2n rows, when there are n statement letters. For 2 statement letters, there are 22 = 2x2 = 4 rows needed. Four rows can consider all the combinations of giving True or False to A and B. We make a diagram like the one below putting A and B on the left, and fill in each row with a different truth possibility for A and B. I want you to do this in a specific way. (So that i. you get all the distinct rows, and ii. we all do it the same way so we can compare). Here goes: 2a. For the first letter, put T in the first half of the rows, and F in the second half of the rows. In our case of A, B we put 2 Ts in the first two rows of the first column and then 2 Fs. Like this: A B T T F F 2b. Then continue filling in the columns each time halving the number of Ts and Fs you put in compared to the column before. Until you get to T, F, T, F, etc. For A, B. A T T F F B T F T F 3. Write the sub-sentences of your main sentence along the top row to the right. Make columns for each sub-sentence until you get to the final connective for that sentence. This method lists the subsentences in increasing complexity from left to right. Recall we are interested in this argument: A -> B ~A Therefore, ~B A B T T F F A-> B ~A ~B T F T F Notice that I put premise 1 in the first column, then premise 2 and then the conclusion. 4. Apply the definitions of the connectives to each column, filling them in order of the smallest compounds, and then building up from the simpler to the more complex. For examples with arguments, try to do this sentence by sentence from premise to conclusion. a. In our example, all the compounds are simple, that is why we used the order of the premises to conclusion. First fill in the column for A -> B, consulting the -> truth-table as your guide. A T T F F B T F T F A-> B T F T T ~A ~B b. Now fill in the next premise, using the ~ definition applied to the A column. A B A-> B ~A ~B T T T F T F F F F T T T F F T T c. Now fill in the conclusion ~B, applying ~ to the B column. A B A-> B ~A ~B T T T F F T F F F T F T T T F F F T T T Now you have completed the truth-table for the three sentences in the argument: A -> B ~A Therefore, ~B E. Applying the definition to determine whether the argument is valid, or not valid. The definition of validity says (as above): An argument consisting only of propositional logic sentences is a deductively valid means, every truth possibility that makes the premises of the argument true, also makes the conclusion true. To determine validity: a. Look at the truth-table you have constructed, and identify all the rows where all the premises are true. In this example it is the third row and the fourth row. A T T F F B T F T F Premise 1 A-> B T F T T P2 ~A F F T T Conc ~B F T F T b. Check that for each row, the conclusion is also true. In this example the conclusion is false on row 3 and true on row 4. This means the argument is not valid. Row 3 makes the premises true and the conclusion false. Any argument that is not valid we call invalid. A -> B ~A Therefore, ~B If you have a driver’s license then you passed the driver’s test. You don’t have a driver’s license. Therefore, you did not pass the driver’s test. A Valid Prop Logic Argument Example: A -> B A Therefore, B. Homework: Do the examples in Govier’s text on page 226. Exercise 2 A to practice determining validity with a truth table. I distribute some of those questions, as well as some questions about equivalence in the file Lecture 6 Questions. -The End of Lecture 6-